# What is the purpose of the Carothers equation?

I don't really understand the equation. I feel like it is simply demonstrating the kind of exponentiality characteristic for the step polymerization reactions, but it cannot be useful for actual calculations. I mean - according to the equation a system which has executed, let's say, a polymerization reaction with 99% conversion will have an average degree of polymerization of 100 even if the system has fewer than 100 monomers. Am I missing something?

## 1 Answer

It is an equation useful for describing the behaviour of a large system, not predicting the fate of a few monomers. Polymerization is a stochastic process—if we only have 80 monomers, the resulting polymer molecules will vary wildly in length if we perform the experiment several times (and logically, could not ever have an average degree of polymerization of 100), and the equation could not provide much insight into the results. However, 80 monomers is a very tiny quantity. If we take one of the monomers used to make PET, terephthalic acid, at 166.13 g/mol works out to around $2\times 10^{-23}\ \textrm{kg}$ for 80 molecules, a quantity that is almost impossible to actually measure out so it doesn't really matter that the equation doesn't work. At a more practical scale, with even 1 g of the monomer, we have ~$10^{21}$ monomers and we will see the average behaviour of the system that the equation describes.